Over the last decade, ocean temperature on the U.S. Northeast Continental Shelf (U.S. NES) has warmed faster than the global average and is associated with observed distribution changes of the northern stock of black sea bass ( Centropristis striata ). Mechanistic models based on physiological responses to environmental conditions can improve future habitat suitability projections. We measured maximum, standard metabolic rate, and hypoxia tolerance (S crit ) of the northern adult black sea bass stock to assess performance across the known temperature range of the species. Two methods, chase and swim-flume, were employed to obtain maximum metabolic rate to examine whether the methods varied, and if so, the impact on absolute aerobic scope. A subset of individuals was held at 30°C for one month (30 chronic °C) prior to experiments to test acclimation potential. Absolute aerobic scope (maximum–standard metabolic rate) reached a maximum of 367.21 mgO 2 kg -1 hr -1 at 24.4°C while S crit continued to increase in proportion to standard metabolic rate up to 30°C. The 30 chronic °C group exhibited a significantly lower maximum metabolic rate and absolute aerobic scope in relation to the short-term acclimated group, but standard metabolic rate or S crit were not affected. This suggests a decline in performance of oxygen demand processes (e.g. muscle contraction) beyond 24°C despite maintenance of oxygen supply. The Metabolic Index, calculated from S crit as an estimate of potential aerobic scope, closely matched the measured factorial aerobic scope (maximum / standard metabolic rate) and declined with increasing temperature to a minimum below 3. This may represent a critical threshold value for the species. With temperatures on the U.S. NES projected to increase above 24°C in the next 80-years in the southern portion of the northern stock’s range, it is likely black sea bass range will continue to shift poleward as the ocean continues to warm.

46Over the last decade, ocean temperature in the U.S. Northeast Continental Shelf (U.S. NES) has 47 warmed faster than the global average and is associated with observed distribution changes of the 48 northern stock of black sea bass (Centropristis striata). Mechanistic models based on 49 physiological responses to environmental conditions can improve future habitat suitability 50 projections. We measured maximum, resting metabolic rate, and hypoxia tolerance (S crit ) of the 51 northern adult black sea bass stock to assess performance across the known temperature range of 52 the species. A subset of individuals was held at 30˚C for one month (30 chronic°C ) prior to 53 experiments to test acclimation potential. Absolute aerobic scope (maximum -resting metabolic 54 rate) reached a maximum of 367.21 mgO 2 kg -1 hr -1 at 24.4°C while S crit continued to increase in 55 proportion to resting metabolic rate up to 30°C. The 30 chronic°C group had a significant decrease 56 in maximum metabolic rate and absolute aerobic scope but resting metabolic rate or S crit were not 57 affected. This suggests a decline in performance of oxygen demand processes (e.g. muscle 58 contraction) beyond 24°C despite maintenance of oxygen supply. The Metabolic Index, 59 calculated from S crit as an estimate of potential aerobic scope, closely matched the measured 60 factorial aerobic scope (maximum / resting metabolic rate) and declined with increasing 61 temperature to a minimum below 3. This may represent a critical value for the species. 62Temperature in the U.S. NES is projected to increase above 24°C in the southern portion of the 63 northern stock's range. Therefore, these black sea bass will likely continue to shift north as the 64 ocean continues to warm. 65 66 67 likely continue to expand northward as the U.S. NES continues to warm. 132 133 Materials and Methods 134 135

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We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hörmander and Rådström. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.